The query concerning the Euler-Poincare formula’s generalizations - MathOverflow

The query concerning the Euler-Poincare formula’s generalizations

2012-05-30T10:58:01Z

Euler's equation for polyhedral, Euler's polyhedral formula, V – E + F = 2, where V, E, and F, are the number of points, edges and faces, was discovered by Leonhard Euler in 1752. However, the basic idea - not the equation - was revealed much earlier by Descartes & others, and later generalized by Lhuilier, as follow: V – E + F = 2 – 2g, where g is genus, the number of holes or handles. Later on, Schläfli and Poincare also generalized the formula to the higher dimensional n-polytopes. We talk about Euler-Poincare formula and Euler-Poincare characteristic (X) for combinatorial cell complexes or polyhedral solids: X = N1 – N2 + N3 – N4 + … +/- Nk, where k is the dimension of the complex Nk and X = 2, if k is odd, or 0, if k is even.

My question is: Are there any other/later generalizations of this Euler-Poincare characteristic of a cell complex?

Answer by Joseph O'Rourke for The query concerning the Euler-Poincare formula’s generalizations

2012-05-30T12:21:58Z

The Wikipedia article discusses (and provides some references for) several generalizations:

To a chain complex, when the Euler characteristic is the alternating sum of the ranks of the homology groups of the chain complex.
To a sheaf on a projective scheme.
To an orbifold, which may have a fractional Euler characteristic.
To a bounded finite poset.
To a finite group or monoid.

Answer by Jorma Kyppö for The query concerning the Euler-Poincare formula’s generalizations

2012-05-30T20:29:30Z

Thank you for your answer. However, I did already study this interesting Wikipedia article. Probably I didn't focus my question enough (first time in mathoverflow). I'm mostly interested about the generalizations, where also the value of genus is aproximated.